An Exponential Mixing Condition for Quantum Channels
Abdessatar Souissi, Abdessatar Barhoumi
TL;DR
This paper develops a quantitative criterion for mixing in quantum channels via the quantum Markov-Dobrushin inequality, showing that a positive Markov-Dobrushin constant $\kappa_{\mathcal{M}}$ guarantees exponential mixing with rate $\theta_{\mathcal{M}} = -\ln(1-\mathrm{Tr}(\kappa_{\mathcal{M}}))$. It proves a contraction bound in the total-variation-like norm and extends the result to convex mixtures with the completely depolarizing channel, $\mathcal{M}_{\alpha}=\alpha\mathcal{M}+(1-\alpha)\Omega$, which are exponentially mixing with $\theta_{\mathcal{M}_{\alpha}} \ge -\ln(\alpha)$. The paper also identifies limitations, showing unistochastic channels are not mixing and characterizes ergodicity for mixed-unitary channels averaged over finite groups, with ergodicity precisely when the group-average channel equals $\Omega$. A detailed qubit example demonstrates mixing behavior and convergence rates, illustrating the practical impact for quantum information tasks. These results illuminate how Markov-Dobrushin-type contractions govern long-time behavior of quantum dynamics and offer avenues for applications in quantum algorithms and walks.
Abstract
Quantum channels, pivotal in information processing, describe transformations within quantum systems and enable secure communication and error correction. Ergodic and mixing properties elucidate their behavior. In this paper, we establish a sufficient condition for mixing based on a quantum Markov-Dobrushin inequality. We prove that if the Markov-Dobrushin constant of a quantum channel exceeds zero, it exhibits exponential mixing behavior. We explore limitations of some quantum channels, demonstrating that unistochastic channels are not mixing. Additionally, we analyze ergodicity of a class of mixed-unitary channels associated with finite groups of unitary operators. Finally, we apply our results to the qubit depolarizing channel.